Maximum principle for fully nonlinear equations via the iterated comparison function method

Shigeaki Koike, Andrzej Świȩch

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Abstract

We present various versions of generalized Aleksandrov-Bakelman-Pucci (ABP) maximum principle for L p -viscosity solutions of fully nonlinear second-order elliptic and parabolic equations with possibly superlinear-growth gradient terms and unbounded coefficients. We derive the results via the "iterated" comparison function method, which was introduced in our previous paper (Koike and Świȩch in Nonlin. Diff. Eq. Appl. 11, 491-509, 2004) for fully nonlinear elliptic equations. Our results extend those of (Koike and Świȩch in Nonlin. Diff. Eq. Appl. 11, 491-509, 2004) and (Fok in Comm. Partial Diff. Eq. 23(5-6), 967-983) in the elliptic case, and of (Crandall et al. in Indiana Univ. Math. J. 47(4), 1293-1326, 1998; Comm. Partial Diff. Eq. 25, 1997-2053, 2000; Wang in Comm. Pure Appl. Math. 45, 27-76, 1992) and (Crandall and Świȩch in Lecture Notes in Pure and Applied Mathematics, vol. 234. Dekker, New York, 2003) in the parabolic case.

Original languageEnglish
Pages (from-to)461-484
Number of pages24
JournalMathematische Annalen
Volume339
Issue number2
DOIs
Publication statusPublished - 2007 Oct 1

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ASJC Scopus subject areas

  • Mathematics(all)

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